Permissive Field of View

Permissive field of view defines visibility more loosely than other Field of Vision methods. A destination square is visible from a source square if there is any unobstructed line from some point in the source square to some point in the destination square. This means that players and monsters will automatically 'peek' around corners, for example. It also means that field of view is symmetric. That is to say that if a destination square is visible from a source square, then that source square is also visible from the destination square. Some approximation algorithms might lose the property of guaranteed symmetry.

One tricky corner case are literally the corners. There are two questions that must be answered. Are corners of squares valid points in the source and destination squares for determining visibility? And do corners of walls obstruct line of sight? Different algorithms may answer these questions in different ways.

Disadvantages

The most complicated method to understand and implement (and therefore debug).

The player cannot sneak up on monsters more easily.

May ruin play balance of current games.

It is more difficult for the player to withdraw from ranged combat (but easier to stay out of range, see Retreating is safe).

Algorithms

There are a number of articles describing different methods:

Isaac_s_fast_beamcasting_LOS -- An approximate algorithm using 'wide beams' sent out at fixed slopes. The larger the radius, the more beams must be sent out to avoid artifacts.

Mutual_Visibility_Field_Of_View -- Uses the corners of squares to determine visibility. In some cases, it does not precisely capture Permissive Field of View. However, the algorithm guarantees symmetry in the field of view.